Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T21:47:07.309Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Classification of Simple Stably Projectionless C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Shaloub Razak
Shaloub Razak*
Affiliation:
The Fields Institute for Research in Mathematical Sciences 222 College St. Toronto, Ontario M5T 3J1, email: razak@math.utoronto.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that simple stably projectionless ${{\text{C}}^{*}}$-algebras which are inductive limits of certain specified building blocks with trivial $\text{K}$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

Keywords

46L3546L05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 1 , 01 February 2002 , pp. 138 - 224
Copyright
Copyright © Canadian Mathematical Society 2002

References

[Bla] Blackadar, B., A simple C*-algebra with no nontrivial projections. Proc. Amer. Math. Soc. 78 (1980), 504–508.Google Scholar
[Dav] Davidson, K., C*-algebras by Example. Fields Institute Monograph Series 6.Google Scholar
[Dea] Dean, A., A continuous field of projectionless C*-algebras. Preprint.Google Scholar
[Ell1] Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38 (1976), 29–44.Google Scholar
[Ell2] Elliott, G. A., On the classification of C*-algebras of real rank zero. J. Reine Angew. Math. 443 (1993), 179–219.Google Scholar
[Ell3] Elliott, G. A., A classification of certain simple C*-algebras. In: Quantum and Non-Commutative Analysis (eds. H. Araki et al.), Kluwer, Dordrecht, 1993, 373–385.Google Scholar
[Ell4] Elliott, G. A., A classification of certain simple C*-algebras II. J. Ramanujan Math. Soc. 12 (1997), 97–134.Google Scholar
[Ell5] Elliott, G. A., The classification problem for amenable C*-algebras. Proceedings of the International Congress of Mathematicians, Zürich, Switzerland, 1994 (ed. S. J. Chatterji), Birkhäuser Verlag, Basel, 1995, 922–932.Google Scholar
[Ell6] Elliott, G. A., An invariant for simple C*-algebras. In: Invited Papers/Articles sollicités, Canadian Mathematical Society 1945–1995, Volume 3 (eds. J. B. Carrell and R. Murty), Canadian Mathematical Society, Ottawa, 1996, 61–90.Google Scholar
[EG] Elliott, G. A. and Gong, G., On the classification of C*-algebras of real rank zero II. Ann. of Math. 144 (1996), 497–610.Google Scholar
[EGL] Elliott, G. A., Gong, G. and Li, L., On simple inductive limits of matrix algebras over higher dimensional spaces, II. Preprint.Google Scholar
[EGJS] Elliott, G. A., Gong, G., Jiang, X. and Su, H., A classification of simple limits of dimension drop C*-algebras. Fields Institute Communications 13 (1997), 125–143.Google Scholar
[EV] Elliott, G. A. and Villadsen, J., Perforated K0-groups. Canad. J. Math., to appear.Google Scholar
[JS] Jiang, X. and Su, H., On a simple unital projectionless C*-algebra. Preprint.Google Scholar
[Kir] Kirchberg, E., The classification of purely infinite C*-algebras using Kasparov's theory. Preprint.Google Scholar
[KK] Kishimoto, A. and Kumjian, A., Simple stably projectionless C*-algebras arising as crossed products. Canad. J. Math. (5) 48 (1996), 980–996.Google Scholar
[Li] Li, L., Simple inductive limit C*-algebras: Spectra and approximation by interval algebras. J. Reine Angew. Math. 507 (1999), 57–79.Google Scholar
[Lin1] Lin, H., A classification theorem for simple C*-algebras of stable rank one, part I. Preprint.Google Scholar
[Lin2] Lin, H., A classification theorem for simple C*-algebras of stable rank one, part II. Preprint.Google Scholar
[NT] Nielsen, K. E. and Thomsen, K., Limits of circle algebras. Expo. Math. 14 (1996), 17–56.Google Scholar
[Phi] Phillips, N. C., A classification theorem for nuclear purely infinite C*-algebras. Doc. Math. 5 (2000), 49–114.Google Scholar
[Ste] Stevens, I., Hereditary subalgebras of certain simple non real rank zero C*-algebras. Fields Institute Monographs 13 (1999), 207–242.Google Scholar
[T] Thomsen, K., Inductive limits of interval algebras: The tracial state space. Amer. J. Math (3) 116 (1994), 605–620.Google Scholar